Download On the moduli of genus 2 curves over finite fields Atsuki UMEGAKI

On the moduli of genus 2 curves over finite fields
Atsuki UMEGAKI (Waseda Univ., Tokyo )
Abstract:
One of the important problems on the arithmetic of curves is to classify the points on
Mg , the moduli space of curves of genus g, satisfying various prescribed properties
and to study how they are distributed.
In this talk, we consider the arithmetic structure of the moduli space Mg (Fq ) of
curves over a finite field Fq . This is equivalent to study the set of unordered (2g +2)tuple in P1 (Fq ) modulo the action of PGL(2), which is stable under Gal(Fq /Fq ).
Naturally, Mg (Fq ) is divided according to the decomposition of (2g + 2) points into
Gal(Fq /Fq ) -orbits.
We shall discuss the case g = 2. We give, among others, some formulas which
represent the behavior of the numbers of these classes with respect to partitions of
the number 6.
Poncelet’s theorem and genus two curves
with real multiplication
Yukiko. SAKAI (Waseda Univ., Tokyo
Abstract:
In 1936, G. Humbert investigated some relations between Poncelet’s Closure Theorem and algebraic curves of genus two with real multiplication (=RM) for small
discriminants. Later, F.Mestre studied them in connection with division points of
elliptic curves.
In this talk, we disscuss the same subject from different point of view. We give
some concrete description of families of such curves with RM, with discriminant D=5
(resp. D=8), which are obtained as a double cover X of a conic C in P2 associated
with the Poncelet’s n-polygon, for n=5 (resp. n=4).
Our study is based on the explicit computation of the lifting of the Poncelet’s
algebraic correspondence on C to the correspondence onX, showing that the latter
induces a nontrivial endomorphism of Pic0 (X), which is identified with the jacobian
variety of X. This enables us to construct a family of curves whose jacobians are of
GL2-type.
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Elliptic curves over Fq (t) and the Bruhat-Tits tree
Andreas Schweizer (NCTS)
Abstract:
The action of the group GL2 (Fq [t]) on the Bruhat-Tits tree T can be considered as
a function field analog of the action of the modular group SL2 (Z) on the complex
upper half-plane.
We discuss how properties of elliptic curves over Fq (t) can be read off from the
(essentially finite) quotient graphs of T .
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Linear relations of quaternion theta series
and vanishing of L(f, s) at s = 1
Ki-ichiro HASHIMOTO (Waseda Univ., Tokyo )
Abstract:
Let B be a definite quaternion algebra over Q with discriminant q. We consider the
arithmetic of orders O of B. We first give a relation of the type number (= number
of isomorphism classes) of orders O and dimensions S20 (qN )(±±) of weight 2 cusp
forms, with prescribed action of Atkin-Lehner operators.
Then we shall discuss the significance of this relation. When the level of O
is square free, it was known by Gross, and Böcherer, Schulze-Pillot that a linear
relation of certain ternary theta series attached to orders corresponds, throgh the
Shimura lifting, to a primitive cusp form of weight 2 having root number +1, whose
L-functions vanish at s = 1.
Based on numerical computations we propose a new conjecture by which the
vanishing of the L-functions at s = 1 for f ∈ S20 (M ) (M arbitrary), should be
controlled by the linear relations of (quaternary) theta series of various levels qN ,
including the case when L(f, s) has root number −1. This suggests a possibility
(= my dream) of finding 0-cycles on modular curves which give rational points of
infinite order on their jacobian varieties.
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